# Measuring Shape with Topology

@article{Macpherson2010MeasuringSW, title={Measuring Shape with Topology}, author={R. D. Macpherson and Benjamin Schweinhart}, journal={ArXiv}, year={2010}, volume={abs/1011.2258} }

We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and self-avoiding random walks.

## 39 Citations

Persistent Homology and the Upper Box Dimension

- MathematicsDiscret. Comput. Geom.
- 2021

A fractal dimension for a metric space based on the persistent homology of subsets of that space is introduced and hypotheses under which this dimension is comparable to the upper box dimension are exhibited.

Fractality and Topology of Self-Avoiding Walks

- Physics
- 2020

We have analyzed geometric and topological features of self-avoiding walks. We introduce a new kind of walk: the loop-deleted self-avoiding walk (LDSAW) motivated by the interaction of chromatin with…

Fractal dimension estimation with persistent homology: A comparative study

- Computer ScienceCommun. Nonlinear Sci. Numer. Simul.
- 2020

The Persistent Homology of Random Geometric Complexes on Fractals

- MathematicsArXiv
- 2018

We study the asymptotic behavior of the persistent homology of i.i.d. samples from a d-Ahlfors regular measure on a metric space — one that satisfies uniform bounds of the form 1 c r ≤ μ (Br (x)) ≤ c…

Statistical Topology of Embedded Graphs

- Mathematics
- 2015

This thesis explores several methods for their analysis, namely swatches, persistent homology, and the knotting of embedded graphs, and proposes a definition for unknotted embedded graphs by examining the relationship between curvature flow and knottedness in graphs.

Topological data analysis and cosheaves

- Mathematics
- 2014

This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level set persistence is presented using sheaves and…

Cyclic Cohomology Groups of Some Self-similar Sets

- Mathematics, Biology
- 2017

The criteria of the variant of the Young integration on cellular self-similar sets are given, and it is shown that the variant is a cyclic 1-cocycle of the algebra of complex-valued Holder continuous functions on the cellularSelf-Similar sets, suggesting that the cocycle is a variant of currents.

Persistent Homology – State of the art and challenges

- Mathematics
- 2016

A recurring task in mathematics, statistics, and computer science is understanding the connectivity information, or equivalently, the topological properties of a given object. For concreteness, we…

Parallel Computation of Persistent Homology using the Blowup Complex

- Mathematics, Computer ScienceSPAA
- 2015

A parallel algorithm that computes persistent homology, an algebraic descriptor of a filtered topological space, by operating on a spatial decomposition of the domain, as opposed to a decomposition with respect to the filtration.

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